Question: For what value of $k$ does the line represented by the equation $-\frac{1}{2}-2kx = 5y$ contain the point $\left(\frac{1}{4},-6\right)$?
Solution: Since $\left(\frac{1}{4}, -6\right)$ lies on the line, we plug $x = \frac{1}{4}$ and $y = -6$ into the equation to get  \begin{align*}
-\frac{1}{2} - \frac{k}{2} &= 5(-6)\\
\Rightarrow\qquad -1-k = -60\\
\Rightarrow\qquad k=\boxed{59}.
\end{align*}